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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The failure of diamond on a reflecting stationary set


Authors: Moti Gitik and Assaf Rinot
Journal: Trans. Amer. Math. Soc. 364 (2012), 1771-1795
MSC (2010): Primary 03E35; Secondary 03E04, 03E05
Published electronically: November 17, 2011
MathSciNet review: 2869191
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Abstract: 1. It is shown that the failure of $ \diamondsuit _S$, for a set % latex2html id marker 518 $ S\subseteq \aleph _{\omega +1}$ that reflects stationarily often, is consistent with $ {\sf GCH}$ and $ \textup {AP}_{\aleph _\omega }$, relative to the existence of a supercompact cardinal. By a theorem of Shelah, $ {\sf GCH}$ and % latex2html id marker 526 $ \square ^*_\lambda $ entails $ \diamondsuit _S$ for any % latex2html id marker 530 $ S\subseteq \lambda ^+$ that reflects stationarily often.

2. We establish the consistency of existence of a stationary subset of $ [\aleph _{\omega +1}]^\omega $ that cannot be thinned out to a stationary set on which the $ \sup $-function is injective. This answers a question of König, Larson and Yoshinobu in the negative.

3. We prove that the failure of a diamond-like principle introduced by Džamonja and Shelah is equivalent to the failure of Shelah's strong hypothesis.


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Additional Information

Moti Gitik
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Email: gitik@post.tau.ac.il

Assaf Rinot
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Address at time of publication: The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Email: assaf@rinot.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05355-9
PII: S 0002-9947(2011)05355-9
Keywords: Diamond, sap, square, approachability, very good scale, reflection
Received by editor(s): May 20, 2009
Received by editor(s) in revised form: March 29, 2010
Published electronically: November 17, 2011
Additional Notes: This research was supported by the Israel Science Foundation (grant No. 234/08). The authors would like to thank the referee for his comments and corrections.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.